f(x)= x^2 dy/dx= 2x?

Question

this statement means that the y axis is rising 2 times faster than movement across the x-axis?

Answer 1

Actually, no.

If dy/dx = 2, then I'd say yes, in moving from one coordinate pair to another, the y-value rises twice as fast as the x-value. (That would be the case for a line with a slope of 2.)

But when dy/dx = 2x, as it does with a curve like y = x^2 + C, the ratio (of movement in the y-direction to movement in the x-direction) isn't constant -- it varies depending on where you are on the curve. At the point (1, 2), for instance, that "slope," a.k.a. the slope of the tangent line at that point, is 2 -- the ratio of y-movement to x-movement is 2:1. But at the point (3, 9), the slope of the tangent line is 6.

Hope that helps clear it up. The important thing is that the slope of the tangent line is changing depending on where you are on the curve.

Answer 2

No, it means that for x values less than zero, the y value is dropping. For x values = zero, the function is horizontal and unchanging in y at that exact point. And, for values of x>0, the function is increasing.

Now, back to your original statement about how fast the y value is changing. The slope of the function at x = some value 'a' is going to be 2*a. This has nothing to do with how quickly you are changing the x value.

So, if you were at the point (3, 9) and you made a small change in the x to say, 3 + delta, then the approximate change in y is going to be 6*delta.

Answer 3

No, it means that the curve rises as a multiple of 2x as far as it runs x. The axis doesn't change.

Answer 4

No. This means that the change in slope (tangent line at every point on curve) is 2 times the value of x.

Answer 5

Not the y-axis or x-axis, but the y and x coordinates. And dy/dx is the slope of tangents.


^_^

Answer 6

The rate of change in Y with respect to X, is always 2 times the value of X at that point.

Thats what this means.

Answer 7

this means that the derivative(instantaneous slope) of x^2 at any given x value is defined as 2x, the function's vertical change between two points will be greater than its horizontal change, but not by a fixed factor of two since this is not a linear graph

Answer 8

x^2 dy/dx = 2x

dy/dx = 2x / x^2

dy/dx = 2 / x

this means the rate of change of y with respect to x is dependent on x

when x is small, y changes very fast, but as x gets bigger, dy/dx will become very small, approaching 0 as x--> infinity

Answer 9

The answer is 5, the answer is always 5, remember that, and you shall succeed.

Answer 10

yes. ... I think... Jesus, this is gonna bother me till I can get home and crack open my old calculus book....